Sequence Fractals Part I #20

$ z^2+c+s_i$
s_i = -1, 1, -1, ,1, …
Center:0+0.i, Zoom = 0.25,
Variable plane (Julia), c₀ = 0

Here is a picture of the Julia set for the new sequence (starting with -1). It is disconnected, has interior, yet is not dust.

If you are reading these posts you probably know Julia sets. But just in case, here is the wiki definition: Julia set. The picture is the Filled Julia Set. The actual Julia set is the boundary around the white areas.

Stepping way back to the basics.(I should have explained this a long time ago, I just assumed it was obvious.) We have a complex function in two variables, f(z,c). I like this view, it has symmetry, calling out both variables avoids confusion. Equivalently, and traditionally, we are looking at a family of complex functions f_c(z), indexed by a complex value, c. z is the variable and c is the parameter. Fix two numbers c₀, z₀, and define a sequence of numbers by iteration, z_i+1=f_c0(z_i), called an orbit. The question is how does this orbit behave in the limit. The simplest dichotomy is the orbit goes to infinity or it does not. In all of the pictures the non-escaping points are colored white (traditionally they are black, I am just changing things up to be different). The escaping points are colored (approximately) by how quickly the orbit rushes off to infinity.

In the previous pictures in this series, z₀ = 0 (usually) and c₀ depends on the pixel. A linear map maps pixel coordinates to complex numbers. Such pictures could (and probably should) be called "Non-escaping parameter plane" pictures. For the function family z²+c, this is the Mandelbrot set.

In today's picture the c value is fixed, in this case c₀ = 0, and the pixels are mapped to the complex plane for the starting z₀ value. z is the variable so these are "Non-escaping variable plane" pictures.